Abelian Group Codes for Channel Coding and Source Coding

In this paper, we study the asymptotic performance of Abelian group codes for the channel coding problem for arbitrary discrete (finite alphabet) memoryless channels as well as the lossy source coding problem for arbitrary discrete (finite alphabet) memoryless sources. For the channel coding problem, we find the capacity characterized in a single-letter information-theoretic form. This simplifies to the symmetric capacity of the channel when the underlying group is a field. For the source coding problem, we derive the achievable rate-distortion function that is characterized in a single-letter information-theoretic form. When the underlying group is a field, it simplifies to the symmetric rate-distortion function. We give several illustrative examples. Due to the nonsymmetric nature of the sources and channels considered, our analysis uses a synergy of information-theoretic and group-theoretic tools.